Theorem iota4an 5115

Description: Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
iota4an	|- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph )

Proof of Theorem iota4an

Step	Hyp	Ref	Expression
1		iota4 5114	. 2 |- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) )
2		euiotaex 5112	. . . 4 |- ( E! x ( ph /\ ps ) -> ( iota x ( ph /\ ps ) ) e. _V )
3		simpl 108	. . . . 5 |- ( ( ph /\ ps ) -> ph )
4	3	sbcth 2926	. . . 4 |- ( ( iota x ( ph /\ ps ) ) e. _V -> [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph ) )
5	2, 4	syl 14	. . 3 |- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph ) )
6		sbcimg 2954	. . . 4 |- ( ( iota x ( ph /\ ps ) ) e. _V -> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph ) <-> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph ) ) )
7	2, 6	syl 14	. . 3 |- ( E! x ( ph /\ ps ) -> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph ) <-> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph ) ) )
8	5, 7	mpbid 146	. 2 |- ( E! x ( ph /\ ps ) -> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph ) )
9	1, 8	mpd 13	1 |- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1481   E!weu 2000   _Vcvv 2689   [.wsbc 2913   iotacio 5094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-sn 3538  df-pr 3539  df-uni 3745  df-iota 5096

This theorem is referenced by: (None)